Projected Richardson varieties and affine Schubert varieties

Abstract

Let $G$ be a complex quasi-simple algebraic group and $G/P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G/P$. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$. Furthermore, we compare the torus-equivariant cohomology and $K$-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type $A$.

20 pages. To appear in Annales de l'institut fourier

References in corpus (6)

• Total positivity, Grassmannians, and networks
• Positroid varieties I: juggling and geometry
• K-classes of matroids and equivariant localization
• From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice
• On the coherence conjecture of Pappas and Rapoport
• Affine Patches on Positroid Varieties and Affine Pipe Dreams (Thesis)

Cited by in corpus (6)

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• Projected Gromov-Witten varieties in cominuscule spaces
• On the $\mu$-admissible set in the extended affine Weyl groups of $E_6$ and $E_7$